Gaussian trivial ring extensions and fqp-rings

Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqpring. Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1 or ∞. Trivial ring extensions are often used to give either examples or counterexamples of rings. One of the most famous is the chain ringR which is not factor of a valuation domain (see [8, X.6] and [7, Theorem 3.5]). This ring R is the trivial ring extension of a valuation domain D by a non-standard uniserial divisible D-module. This example gives a negative answer to a question posed by Kaplansky. In [14], [15] and [1] there are many results on trivial ring extensions and many examples of such rings. In particular, necessary and sufficient conditions are given for the trivial ring extension of a ring A by an A-module E to be either arithmetical or Gaussian in the following cases: either A is a domain and K is its quotient field, or A is local and K is its residue field, and E is a K-vector space. In our paper more general results are shown. For instance the trivial ring extension R of a ring A by a non-zero A-module E is a chain ring if and only if A is a valuation domain and E a divisible module, and R is Gaussian if and only if A is Gaussian and E verifies aE = aE for each a ∈ A. Complete characterizations of arithmetical trivial ring extensions are given too. But Bezout trivial ring extensions of a ring A are characterized only in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. We also study trivial ring extensions which are fqp-rings. The class of fqprings was introduced in [1] by Abuhlail, Jarrar and Kabbaj. We get a complete characterization of trivial ring extensions which are fqp-ring only in the local case. Each fqp-ring is locally fqp and the converse holds if it is coherent, but this is not generally true. We introduce the class of fqf-rings which satisfy the condition ”each finitely generated ideal is flat modulo its annihilator”, and it is exactly the class of locally fqp-rings. So, trivial fqf-ring extensions are completely characterized. This new class of rings contains strictly the class of fqp-rings. In [6] it is proven that each arithmetical ring has a finitistic weak dimension equal to 0, 1 or 2. We show that any fqf-ring satisfy this property too, and its global weak dimansion is 0, 1 or ∞ as it is shown for fqp-rings in [1]. 2010 Mathematics Subject Classification. 13F30, 13C11, 13E05.